Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour).

Options

A. `84`
B. `360`
C. `504`
D. none of these

Explanation

To solve the problem of arranging 7 green bottles and 8 blue bottles in a row with the condition that exactly 1 pair of green bottles is side by side, we can follow these steps:

### Step 1: Treat the Pair of Green Bottles as a Single Unit
Since we need exactly one pair of green bottles to be side by side, we can treat this pair as a single unit. This means instead of having 7 green bottles, we will consider the pair as one "super bottle." Thus, we have:
- 1 pair of green bottles (treated as 1 unit)
- 5 remaining green bottles (since 7 - 2 = 5)
- 8 blue bottles

So, we now have a total of:
- 1 (pair) + 5 (remaining green) + 8 (blue) = 14 units to arrange.

### Step 2: Calculate the Total Arrangements
Now, we need to arrange these 14 units (1 pair of green + 5 green + 8 blue). The formula for arranging n items where there are groups of identical items is given by:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2!} \]

Where:
- \( n \) is the total number of items,
- \( n_1 \) is the number of identical items of one type,
- \( n_2 \) is the number of identical items of another type.

In our case:
- Total units \( n = 14 \)
- Green bottles \( n_1 = 6 \) (1 pair + 5 remaining)
- Blue bottles \( n_2 = 8 \)

Thus, the total arrangements can be calculated as:

\[ \text{Total arrangements} = \frac{14!}{6! \times 8!} \]

### Step 3: Calculate the Factorials
Now we can calculate the factorials:

\[ 14! = 87178291200 \]
\[ 6! = 720 \]
\[ 8! = 40320 \]

### Step 4: Substitute and Simplify
Now substituting these values into the formula:

\[ \text{Total arrangements} = \frac{87178291200}{720 \times 40320} \]

Calculating the denominator:

\[ 720 \times 40320 = 29030400 \]

Now substituting back:

\[ \text{Total arrangements} = \frac{87178291200}{29030400} = 3003 \]

### Step 5: Consider the Pair of Green Bottles
Since we have treated the pair of green bottles as a single unit, we need to consider that there are 2 ways to arrange the pair of green bottles (since they are identical). Thus, we multiply the total arrangements by 2:

\[ \text{Final arrangements} = 3003 \times 2 = 6006 \]

### Final Answer
The number of ways to arrange 7 green bottles and 8 blue bottles in a row such that exactly 1 pair of green bottles is side by side is **6006**.

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